Linear and Nonlinear Elliptic and Parabolic Partial Differential Equations

Lecturer

Dr Yann Bernard, Monash University

Synopsis

Elliptic and parabolic partial differential equations are central to the mathematical modelling of physical, geometric, and biological systems. Elliptic equations, such as the Poisson equation, describe steady-state or equilibrium configurations with applications in electrostatics, fluid mechanics, and differential geometry, inter alia. Solutions to elliptic PDEs typically exhibit strong regularity and obey maximum principles, making them analytically tractable and structurally very rich. Parabolic equations, such as the heat equation, govern time-evolving processes involving diffusion and dissipation. These include heat flow, chemical diffusion, population dynamics, and geometric flows like the Ricci and mean curvature flows. Parabolic PDEs tend to smooth out irregularities in initial data over time, revealing long-term behaviour and stability.

The analytical study of partial differential equations was rigorously initiated in the early-to-mid-XXth century and has tremendously grown since then. This introductory course will cover the core results of the theory, and will be theoretical in nature: we will address such questions as existence and uniqueness, but also questions regarding the regularity of solutions, culminating in the resolution of Hilbert’s XIXth problem: are solutions to regular variational problems always analytic?

Course Overview

Week 1: Foundations and Classical Theory of Elliptic Equations

  • Introduction
  • Harmonic functions
  • Perron’s method

Week 2: Weak Solutions and Sobolev Spaces 

  • General second-order linear elliptic equations
  • Sobolev spaces
  • Embedding theorems

Week 3: Linear Parabolic Equations and Schauder Theory

  • Weak solutions of the Dirichlet problem
  • Weak maximum principle
  • Regularity of weak solutions

Week 4: Classical Nonlinear Elliptic and Parabolic Equations

  • The direct method of Calculus of variation
  • Existence/regularity of minimisers
  • Theorem of DeGiorgi-Nash-Moser
  • Parabolic equations

Prerequisites

  • Students should have a good command of multivariable Calculus and be comfortable with real analysis. Fluency in the rudiments of functional analysis and measure theory is definitely useful (in particular the Hahn-Banach theorem, Hilbert spaces, Lebesgue spaces) but not essential.

Assessment

  • 2x take-home assignments – 25% each
  • Take home examination – 50%

Resources/pre-reading

  • Most of the material covered is found in the classical texts:
    • Elliptic partial differential equations of second order by D. Gilbarg and N. Trudinger, in Partial differential equations by L.C. Evans, and
    • Elliptic partial differential equations by Q. Han and F.-H. Lin.
  • A good pre-reading is the concise overview of partial differential equations given in sections 1.1-1.3 in Evans’ book (above).

Not sure if you should sign up for this course?

Check back for pre-enrolment QUIZ details so you can self-evaluate and get a measure of the key foundational knowledge required.

Dr Yann Bernard, Monash University

Yann Bernard completed his doctoral studies at the University of Michigan in 2006. He was a research fellow/postdoc at ETH in Zurich and at the University of Freiburg, before joining the School of Mathematics at Monash University as a senior lecturer in 2015. He currently serves as director of postgraduate studies. His research deals with the study of solutions to conformally invariant variational problems arising in geometric analysis; with applications to elasticity theory, general relativity, and the AdS/CFT correspondence, among others.